"Function y=Asin(ωx+φ)" Trigonometric function PPT

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"Function y=Asin(ωx+φ)" Trigonometric function PPT

Part One: Learning Objectives

1. Understand the influence of parameters A, ω, φ on the image of function y = Asin (ωx + φ); be able to transform the image of y = sin x to obtain the image of y = Asin (ωx + φ), x∈R. (difficulty)

2. Be able to determine its analytical formula based on the partial image of y=Asin(ωx+φ). (emphasis)

3. Determine the value of φ when finding the analytical expression of a function. (easy to make mistakes)

core competencies

1. Cultivate intuitive imagination through the transformation of function images.

2. Use the image of the function to find the analytical expression and improve the mathematical operation literacy.

Function y=Asin(ωx+φ)PPT, the second part of the content: independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. The influence of φ on the image of y=sin(x+φ), x∈R

2. The influence of ω (ω>0) on the image of y=sin (ωx+φ)

3. The influence of A (A>0) on the image of y=Asin(ωx+φ)

First try

1. The analytical formula of the image obtained by shifting the image of the function y=sin x to the left by π3 unit lengths is ()

A． y＝sin x－π3 B． y＝sinx＋π3

C． y=sinx-π3 D． y＝sinx＋π3

2. In order to get the image of the function y=4sin12x-π6, x∈R, just put all the points on the image of the function y=4sinx-π6, x∈R ()

A． The abscissa is stretched to twice its original size, and the ordinate remains unchanged.

B. The abscissa is shortened to 12 times its original size, and the ordinate remains unchanged.

C． The ordinate is stretched to twice its original size, and the abscissa remains unchanged.

D. The ordinate is shortened to 12 times its original size, and the abscissa remains unchanged.

3. The maximum value of the function y=Asin(ωx+φ)+1(A>0, ω>0) is 5, then A=________.

Function y=Asin(ωx+φ)PPT, the third part of the content: cooperative exploration to improve literacy

Transformation between graphs of trigonometric functions

[Example 1] (1) Translate the image of the function y=2cos2x+π3 to the left by π3 unit lengths, and then translate it downward by 3 unit lengths. Then the analytical formula of the resulting image is _________.

(2) How to transform the image of y=sin x to obtain the image of the function y=2sin2x+π4+1?

[Ideas Enlightenment] (1) Write the analytical expression according to the rules of left addition and right subtraction; upper addition and lower subtraction.

(2) Method 1: y = sin x → ordinate expansion and contraction → abscissa expansion and contraction → upward translation.

Method 2: Translate left and right → telescope on the abscissa → telescope on the ordinate → translate up and down.

regular method

From the image of y=sin

(1)y=sin x����→Phase transformation y=sin(x+φ)����→Period transformation y=sin(ωx+φ)

����→Amplitude transformation y=Asin(ωx+φ).

(2)y=sin x����→Period transformation y=sin ωx����→Phase transformation y=sinωx＋φω=sin(ωx＋φ)����→Amplitude transformation y=Asin(ωx＋φ).

Reminder: The transformation order of the two approaches is different, and the amount of transformation is also different: (1) It is phase transformation first and then period transformation, with a translation of |φ| units. (2) It is a period transformation first and then a phase transformation, with a translation of |φ|ω units. This is a very error-prone place and should be paid special attention to.

Find the analytical expression of the graph of a given function

[Example 2] (1) It is known that the partial image of the function f(x)=Acos(ωx+φ)+BA>0, ω>0, |φ|<π2 is as shown in the figure, then the function f ( The analytical formula of x) is ()

A． y=2cosx2-π4+4 B. y=2cosx2+π4+4

C． y＝4cosx2－π4＋2 D. y=4cosx2+π4+2

(2) In the function f(x)=Asin(ωx+φ), A>0, ω>0, |φ|<π2, and the image is as shown in the figure, find its analytical formula.

[Idea Tips] Find A and B from the maximum (small) value, find ω from the period, and solve the equation from the special point coordinates to find φ.

regular method

The key to determining the analytical formula of the function y=Asinωx+φ is the determination of φ. Commonly used methods are:

1 Substitution method: Substitute a known point on the image into At this time, A and ω are known or substitute into the intersection point of the image and the x-axis to solve the problemAt this time, pay attention to whether the intersection point is on the rising interval or the falling interval.

2 Five-point method: When determining the value of φ, it is often necessary to find the first zero point in the "five-point method" - φω, 0 as a breakthrough. The value of ωx + φ of the "five-point" is as follows: The "first point" is the figure The intersection point  with the x-axis when the image rises is ωx + φ = 0; the “second point”, which is the “peak point” of the image, is ωx + φ = π2; and the “third point”, which is the intersection point with the x-axis when the image descends  is ωx+φ=π; the “fourth point”, which is the “valley point” of the image, is ωx+φ=3π2; and the “fifth point” is ωx+φ=2π.

Comprehensive application of images and properties of trigonometric functions

[Inquiry Questions]

1. How to find the symmetry axis equation of the functions y=Asin(ωx+φ) and y=Acos(ωx+φ)?

Tip: Like sine curves and cosine curves, the symmetry axis of the graph of functions y=Asin(ωx+φ) and y=Acos(ωx+φ) passes through the maximum point of the function graph and is perpendicular to the x-axis.

2. How to find the symmetry center of the functions y=Asin(ωx+φ) and y=Acos(ωx+φ)?

Tip: Like sine curves and cosine curves, the symmetry center of the graphs of functions y=Asin(ωx+φ) and y=Acos(ωx+φ) is the intersection point of the function graph and the x-axis.

How to find the symmetry center of function y=Asin(ωx+φ): Let sin(ωx+φ)=0, and get ωx+φ=kπ(k∈Z), then x=kπ-φω(k∈Z), so the function y=Asin(ωx+φ) The image is centrally symmetric about the point kπ-φω, 0 (k∈Z);

Class summary

1. Accurately understand "image transformation method"

(1) The image transformation from y=sin x to y=sin (x+φ) is called phase transformation, and the transformation from y=sin x to y=sin ωx is called periodic transformation; from y=sin =The transformation of Asin x image is called amplitude transformation.

(2) From the image of y=sin : ① It is phase transformation first and then period transformation, with a translation of |φ| units. ② It is to perform periodic transformation first and then phase transformation, with a translation of |φ|ω units. This is a very error-prone place and should be paid special attention to.

(3) Similarly, the image of y=Acos (ωx+φ) (A>0, ω>0) can also be obtained by transforming the image of y=cos x.

2. The key to determining the analytical formula from the image properties or partial images of y=Asin (ωx+φ) is to determine the parameters A, ω, φ. The basic method is to use the undetermined coefficient method to solve the problem based on observing the image.

Function y=Asin(ωx+φ)PPT, part 4 content: Complying with the standard in class and solidifying the double base

1. Thinking and analysis

(1) The analytical formula of the image obtained by shifting the image of y=sin 3x π4 units to the left is y=sin3x+π4.()

(2) When the abscissas of all points on the image of y=sin x are doubled, the analytical formula of the resulting image is y=sin 2x.()

(3) The ordinates of all points on the image of y=sin x are doubled. The analytical formula of the resulting image is y=12 sin x. ()

2. The ordinate of each point on the function y=cos

3. There are two main processes for transforming the image of y=3sin _______ units.

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Update Time: 2024-10-06

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